How to perform Wilsonian RG for the $O(N)$ nonlinear sigma model?

Question

I am interested in deriving the $\beta$ function and anomalous dimension $\gamma$ of the $O(N)$ nonlinear $\sigma$ model, in particular defined by the action $S = \frac{1}{2g} \int d^2 x (\partial_{\mu}\vec{n}(x))^2$ with the constraint $\vec{n}(x)^2 = 1$. I tried to follow Fradkin's derivation in his QFT book, but I feel it is unnecessarily complicated and there seem to be some errors(?). In particular, I am only interested in the dominant contributions to the RG flow and not a proof of the renormalizability of the model. I would prefer a Wilsonian type derivation where modes are integrated out in a momentum shell to derive the flow.

In particular, following Fradkin's method of using the constraint to integrate out one component of $\vec{n}(x) = (\vec{\pi}(x), \sigma(x))$, getting a contact term in the Lagrangian coming from the functional integration measure, I map the partition function to

$$ Z = \int D\vec{\pi} e^{-S_{eff}[\vec{\pi}]} $$

with the effective action given by

$$ S_{eff}[\vec{\pi}] = \int d^2 x \frac{1}{2g} [(\partial_{\mu}\vec{\pi})^2 + (\partial_{\mu}\sqrt{1-\vec{\pi}^2})^2] + \frac{1}{2a^d}\ln(1-\vec{\pi}^2) $$

where $a\sim \Lambda^{-1}$ is the lattice constant and $\Lambda$ is the UV cutoff. The lowest order interaction takes the form $\frac{1}{2g}(\vec{\pi} \cdot \partial_{\mu}\pi)^2$, which should contribute to the self-energy at 1-loop order, giving the dominant contribution to the renormalization of $g$. However calculating the result of the bubble diagram, I cannot see how to reproduce the correct result of $\beta(g) = \frac{1}{2\pi} (N-2)g^2$. In fact it seems the indices of $\pi^a$ within the loop are fixed by the external lines, and so there should be no contribution of $N$ at all at one loop order. Also, how is the anomalous dimension extracted in this theory? I am used to thinking of the anomalous dimension as arising from a field strength renormalization, i.e. the kinetic term becomes $Z (\partial_{\mu} \phi)^2$ with $Z$ flowing with $\Lambda$, leading to an anomalous dimension. However g basically plays the role of renormalizing the field strength in this case, as it appears in the prefactor of the kinetic term.

In Fradkin's derivation, he uses an auxiliary symmetry breaking source term coupling to $\sqrt{1-\vec{\pi}^2}$ to derive $\beta(g)$ and $\gamma$. I do not understand why or if this is necessary. Is the theory not well defined without the presence of a source term? Is it possible to reproduce the results without a source term?

Answer 1

In the absence of the symmetry breaking term, the massless Goldstone modes make the diagrams infra-red divergent. You can avoid this by only computing correlators that are $O(n)$ symmetric such as $\langle{\bf n}(x)\cdot {\bf n}(x')\rangle$. The calculation is done in S. Elitzur's paper Nucl. Phys., B212 (1983), p. 501

Answer 2

I found that the relevant derivation is presented in much more detail in Zinn-Justin's book, chapter 15. The crux is the parameterization $\vec{n}(x) = (\vec{\pi}(x), \sqrt{1-\vec{\pi}(x)^2})$ implicitly assumes that the fluctuations $\vec{\pi}(x)$ are small, and therefore that we are in the SSB phase of the $O(N)$ symmetry, and that the fields $\vec{\pi}(x)$ are the massless Goldstone modes of this SSB. In other words, our whole expansion relies on $\langle\sigma(x)\rangle = \langle\sqrt{1-\vec{\pi}(x)^2}\rangle \sim 1$. However in $d=2$ dimensions, the 1-loop contribution to this expectation value is actually divergent, destroying the order as required by the Mermin-Wagner theorem. The source term then acts as an IR regulator, explicitly breaking the $O(N)$ symmetry and giving mass to the Goldstone modes so that one can still organize a perturbative expansion.